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@ -213,6 +213,7 @@ Insulators</a></li>
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<li><a href="#quantum-spin-liquids"
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id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
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<li><a href="#outline" id="toc-outline">Outline</a></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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</ul>
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{% endcapture %}
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@ -230,6 +231,7 @@ Insulators</a></li>
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<li><a href="#quantum-spin-liquids"
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id="toc-quantum-spin-liquids">Quantum Spin Liquids</a></li>
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<li><a href="#outline" id="toc-outline">Outline</a></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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</ul>
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</nav>
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-->
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@ -266,9 +268,10 @@ that of the individual objects.</p>
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data-short-caption="A murmuration of Starlings" style="width:100.0%"
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alt="Figure 1: A murmuration of starlings. Dorset, UK. Credit Tanya Hart, “Studland Starlings”, 2017, CC BY-SA 3.0" />
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<figcaption aria-hidden="true"><span>Figure 1:</span> A murmuration of
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starlings. Dorset, UK. Credit <a href="twitter.com/arripay">Tanya
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Hart</a>, “Studland Starlings”, 2017, <a
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href="creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA
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starlings. Dorset, UK. Credit <a
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href="https://twitter.com/arripay">Tanya Hart</a>, “Studland Starlings”,
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2017, <a
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href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">CC BY-SA
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3.0</a></figcaption>
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</figure>
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</div>
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@ -356,7 +359,7 @@ methods.</p>
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<figure>
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<img src="/assets/thesis/intro_chapter/venn_diagram.svg"
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data-short-caption="Interacting Quantum Many Body Systems Venn Diagram"
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style="width:57.0%"
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style="width:100.0%"
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alt="Figure 2: Three key adjectives. Many Body, the fact of describing systems in the limit of large numbers of particles. Quantum, objects whose behaviour requires quantum mechanics to describe accurately. Interacting, the constituent particles of the system affect one another via forces, either directly or indirectly. When taken together, these three properties can give rise to what are called strongly correlated materials." />
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<figcaption aria-hidden="true"><span>Figure 2:</span> Three key
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adjectives. Many Body, the fact of describing systems in the limit of
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@ -531,14 +534,14 @@ href="#ref-law1TTaS2QuantumSpin2017" role="doc-biblioref">26</a>,<a
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href="#ref-ribakGaplessExcitationsGround2017"
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role="doc-biblioref">27</a>]</span>, giving rise to what is known as a
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quantum spin liquid (QSL) state.</p>
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<p>Landau theory characterises phases of matter as inextricably linked
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to the emergence of long range order via a spontaneously broken
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symmetry. The fractional quantum Hall (FQH) state, discovered in the
|
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1980s is an explicit example of an electronic system that falls outside
|
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of the Landau paradigm. FQH systems exhibit fractionalised excitations
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linked to their ground state having long range entanglement and
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non-trivial topological properties <span class="citation"
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data-cites="broholmQuantumSpinLiquids2020"> [<a
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<p>Landau-Ginzburg-Wilson theory characterises phases of matter as
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inextricably linked to the emergence of long range order via a
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spontaneously broken symmetry. The fractional quantum Hall (FQH) state,
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discovered in the 1980s is an explicit example of an electronic system
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that falls outside of the Landau-Ginzburg-Wilson paradigm. FQH systems
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exhibit fractionalised excitations linked to their ground state having
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long range entanglement and non-trivial topological properties <span
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class="citation" data-cites="broholmQuantumSpinLiquids2020"> [<a
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href="#ref-broholmQuantumSpinLiquids2020"
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role="doc-biblioref">40</a>]</span>. Quantum spin liquids are the
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analogous phase of matter for spin systems. Remarkably the existence of
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@ -595,33 +598,40 @@ problem via a mapping to Majorana fermions which yields an extensive
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number of static <span class="math inline">\(\mathbb Z_2\)</span> fluxes
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tied to an emergent gauge field. The model is remarkable not only for
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its QSL ground state, it supports a rich phase diagram hosting gapless,
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Abelian and non-Abelian phases and a finite temperature phase transition
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to a thermal metal state <span class="citation"
|
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Abelian and non-Abelian phases <span class="citation"
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data-cites="knolleDynamicsFractionalizationQuantum2015"> [<a
|
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href="#ref-knolleDynamicsFractionalizationQuantum2015"
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role="doc-biblioref">51</a>]</span> and a finite temperature phase
|
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transition to a thermal metal state <span class="citation"
|
||||
data-cites="selfThermallyInducedMetallic2019"> [<a
|
||||
href="#ref-selfThermallyInducedMetallic2019"
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role="doc-biblioref">51</a>]</span>. It has also been proposed that it
|
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could be used to support topological quantum computing <span
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class="citation"
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role="doc-biblioref">52</a>]</span>. It been proposed that its
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non-Abelian excitations could be used to support robust topological
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quantum computing [<span class="citation"
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data-cites="kitaev_fault-tolerant_2003"> [<a
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||||
href="#ref-kitaev_fault-tolerant_2003"
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role="doc-biblioref">53</a>]</span>; <span class="citation"
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data-cites="freedmanTopologicalQuantumComputation2003"> [<a
|
||||
href="#ref-freedmanTopologicalQuantumComputation2003"
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||||
role="doc-biblioref">52</a>]</span>.</p>
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role="doc-biblioref">54</a>]</span>;
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nayakNonAbelianAnyonsTopological2008].</p>
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<p>It is by now understood that the Kitaev model on any tri-coordinated
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<span class="math inline">\(z=3\)</span> graph has conserved plaquette
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operators and local symmetries <span class="citation"
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data-cites="Baskaran2007 Baskaran2008"> [<a href="#ref-Baskaran2007"
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role="doc-biblioref">53</a>,<a href="#ref-Baskaran2008"
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||||
role="doc-biblioref">54</a>]</span> which allow a mapping onto effective
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role="doc-biblioref">55</a>,<a href="#ref-Baskaran2008"
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role="doc-biblioref">56</a>]</span> which allow a mapping onto effective
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free Majorana fermion problems in a background of static <span
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class="math inline">\(\mathbb Z_2\)</span> fluxes <span class="citation"
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data-cites="Nussinov2009 OBrienPRB2016 yaoExactChiralSpin2007 hermanns2015weyl"> [<a
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||||
href="#ref-Nussinov2009" role="doc-biblioref">55</a>–<a
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href="#ref-hermanns2015weyl" role="doc-biblioref">58</a>]</span>.
|
||||
href="#ref-Nussinov2009" role="doc-biblioref">57</a>–<a
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||||
href="#ref-hermanns2015weyl" role="doc-biblioref">60</a>]</span>.
|
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However, depending on lattice symmetries, finding the ground state flux
|
||||
sector and understanding the QSL properties can still be
|
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challenging <span class="citation"
|
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data-cites="eschmann2019thermodynamics Peri2020"> [<a
|
||||
href="#ref-eschmann2019thermodynamics" role="doc-biblioref">59</a>,<a
|
||||
href="#ref-Peri2020" role="doc-biblioref">60</a>]</span>.</p>
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||||
href="#ref-eschmann2019thermodynamics" role="doc-biblioref">61</a>,<a
|
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href="#ref-Peri2020" role="doc-biblioref">62</a>]</span>.</p>
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<p><strong>paragraph about amorphous lattices</strong></p>
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<p>In Chapter 4 I will introduce a soluble chiral amorphous quantum spin
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liquid by extending the Kitaev honeycomb model to random lattices with
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@ -640,6 +650,7 @@ localisation.</p>
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<p>In Chapter 3 I introduce the Long Range Falikov-Kimball Model in
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greater detail. I will present results that. Chapter 4 focusses on the
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Amorphous Kitaev Model.</p>
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<h1 class="unnumbered" id="bibliography">Bibliography</h1>
|
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<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
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<div id="ref-king2012murmurations" class="csl-entry"
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role="doc-biblioentry">
|
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@ -1032,32 +1043,48 @@ Kitaev, <em><a href="https://doi.org/10.1016/j.aop.2005.10.005">Anyons
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in an Exactly Solved Model and Beyond</a></em>, Annals of Physics
|
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<strong>321</strong>, 2 (2006).</div>
|
||||
</div>
|
||||
<div id="ref-knolleDynamicsFractionalizationQuantum2015"
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||||
class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[51] </div><div class="csl-right-inline">J.
|
||||
Knolle, D. L. Kovrizhin, J. T. Chalker, and R. Moessner, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.92.115127">Dynamics of
|
||||
Fractionalization in Quantum Spin Liquids</a></em>, Phys. Rev. B
|
||||
<strong>92</strong>, 115127 (2015).</div>
|
||||
</div>
|
||||
<div id="ref-selfThermallyInducedMetallic2019" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[51] </div><div class="csl-right-inline">C.
|
||||
<div class="csl-left-margin">[52] </div><div class="csl-right-inline">C.
|
||||
N. Self, J. Knolle, S. Iblisdir, and J. K. Pachos, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.99.045142">Thermally Induced
|
||||
Metallic Phase in a Gapped Quantum Spin Liquid - a Monte Carlo Study of
|
||||
the Kitaev Model with Parity Projection</a></em>, Phys. Rev. B
|
||||
<strong>99</strong>, 045142 (2019).</div>
|
||||
</div>
|
||||
<div id="ref-kitaev_fault-tolerant_2003" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[53] </div><div class="csl-right-inline">A.
|
||||
Yu. Kitaev, <em><a
|
||||
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-Tolerant
|
||||
Quantum Computation by Anyons</a></em>, Annals of Physics
|
||||
<strong>303</strong>, 2 (2003).</div>
|
||||
</div>
|
||||
<div id="ref-freedmanTopologicalQuantumComputation2003"
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||||
class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[52] </div><div class="csl-right-inline">M.
|
||||
<div class="csl-left-margin">[54] </div><div class="csl-right-inline">M.
|
||||
Freedman, A. Kitaev, M. Larsen, and Z. Wang, <em><a
|
||||
href="https://doi.org/10.1090/S0273-0979-02-00964-3">Topological Quantum
|
||||
Computation</a></em>, Bull. Amer. Math. Soc. <strong>40</strong>, 31
|
||||
(2003).</div>
|
||||
</div>
|
||||
<div id="ref-Baskaran2007" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[53] </div><div class="csl-right-inline">G.
|
||||
<div class="csl-left-margin">[55] </div><div class="csl-right-inline">G.
|
||||
Baskaran, S. Mandal, and R. Shankar, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevLett.98.247201">Exact Results for
|
||||
Spin Dynamics and Fractionalization in the Kitaev Model</a></em>, Phys.
|
||||
Rev. Lett. <strong>98</strong>, 247201 (2007).</div>
|
||||
</div>
|
||||
<div id="ref-Baskaran2008" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[54] </div><div class="csl-right-inline">G.
|
||||
<div class="csl-left-margin">[56] </div><div class="csl-right-inline">G.
|
||||
Baskaran, D. Sen, and R. Shankar, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.78.115116">Spin-S Kitaev Model:
|
||||
Classical Ground States, Order from Disorder, and Exact Correlation
|
||||
@ -1065,14 +1092,14 @@ Functions</a></em>, Phys. Rev. B <strong>78</strong>, 115116
|
||||
(2008).</div>
|
||||
</div>
|
||||
<div id="ref-Nussinov2009" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[55] </div><div class="csl-right-inline">Z.
|
||||
<div class="csl-left-margin">[57] </div><div class="csl-right-inline">Z.
|
||||
Nussinov and G. Ortiz, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.79.214440">Bond Algebras and
|
||||
Exact Solvability of Hamiltonians: Spin S=½ Multilayer Systems</a></em>,
|
||||
Physical Review B <strong>79</strong>, 214440 (2009).</div>
|
||||
</div>
|
||||
<div id="ref-OBrienPRB2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[56] </div><div class="csl-right-inline">K.
|
||||
<div class="csl-left-margin">[58] </div><div class="csl-right-inline">K.
|
||||
O’Brien, M. Hermanns, and S. Trebst, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.93.085101">Classification of
|
||||
Gapless Z₂ Spin Liquids in Three-Dimensional Kitaev Models</a></em>,
|
||||
@ -1080,27 +1107,27 @@ Phys. Rev. B <strong>93</strong>, 085101 (2016).</div>
|
||||
</div>
|
||||
<div id="ref-yaoExactChiralSpin2007" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[57] </div><div class="csl-right-inline">H.
|
||||
<div class="csl-left-margin">[59] </div><div class="csl-right-inline">H.
|
||||
Yao and S. A. Kivelson, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevLett.99.247203">An Exact Chiral
|
||||
Spin Liquid with Non-Abelian Anyons</a></em>, Phys. Rev. Lett.
|
||||
<strong>99</strong>, 247203 (2007).</div>
|
||||
</div>
|
||||
<div id="ref-hermanns2015weyl" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[58] </div><div class="csl-right-inline">M.
|
||||
<div class="csl-left-margin">[60] </div><div class="csl-right-inline">M.
|
||||
Hermanns, K. O’Brien, and S. Trebst, <em>Weyl Spin Liquids</em>,
|
||||
Physical Review Letters <strong>114</strong>, 157202 (2015).</div>
|
||||
</div>
|
||||
<div id="ref-eschmann2019thermodynamics" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[59] </div><div class="csl-right-inline">T.
|
||||
<div class="csl-left-margin">[61] </div><div class="csl-right-inline">T.
|
||||
Eschmann, P. A. Mishchenko, T. A. Bojesen, Y. Kato, M. Hermanns, Y.
|
||||
Motome, and S. Trebst, <em>Thermodynamics of a Gauge-Frustrated Kitaev
|
||||
Spin Liquid</em>, Physical Review Research <strong>1</strong>, 032011(R)
|
||||
(2019).</div>
|
||||
</div>
|
||||
<div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[60] </div><div class="csl-right-inline">V.
|
||||
<div class="csl-left-margin">[62] </div><div class="csl-right-inline">V.
|
||||
Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R.
|
||||
Moessner, and R. Thomale, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.101.041114">Non-Abelian Chiral
|
||||
|
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@ -282,6 +282,7 @@ Chern number</a></li>
|
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<li><a href="#phase-diagram" id="toc-phase-diagram">Phase
|
||||
Diagram</a></li>
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</ul></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
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</ul>
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{% endcapture %}
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@ -305,10 +306,13 @@ Chern number</a></li>
|
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<li><a href="#phase-diagram" id="toc-phase-diagram">Phase
|
||||
Diagram</a></li>
|
||||
</ul></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
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</nav>
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-->
|
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<h1 id="the-kitaev-honeycomb-model">The Kitaev Honeycomb Model</h1>
|
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<p><strong>papers</strong> Jos on dynamics
|
||||
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.115127</p>
|
||||
<p><strong>intro</strong> - strong spin orbit coupling leads to
|
||||
anisotropic spin exchange (as opposed to isotropic exchange like the
|
||||
Heisenberg model) - geometrical frustration leads to QSL ground state
|
||||
@ -360,6 +364,7 @@ Chern number</h2>
|
||||
<h2 id="phase-diagram">Phase Diagram</h2>
|
||||
<div class="sourceCode" id="cb1"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-kugelJahnTellerEffectMagnetism1982" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
|
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@ -318,6 +318,7 @@ Review</a></li>
|
||||
<li><a href="#the-falikov-kimball-model-1"
|
||||
id="toc-the-falikov-kimball-model-1">The Falikov-Kimball model</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
{% endcapture %}
|
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|
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@ -377,6 +378,7 @@ Review</a></li>
|
||||
<li><a href="#the-falikov-kimball-model-1"
|
||||
id="toc-the-falikov-kimball-model-1">The Falikov-Kimball model</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
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@ -1419,6 +1421,7 @@ electons that can move and those that can’t.</p>
|
||||
<div class="sourceCode" id="cb2"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<p></ij></ij></ij></ij></ij></ij></p>
|
||||
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-hodsonOnedimensionalLongrangeFalikovKimball2021"
|
||||
class="csl-entry" role="doc-biblioentry">
|
||||
|
||||
@ -346,6 +346,7 @@ distribution</a></li>
|
||||
<li><a href="#two-step-trick-2" id="toc-two-step-trick-2">Two Step
|
||||
Trick</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
{% endcapture %}
|
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|
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@ -433,6 +434,7 @@ distribution</a></li>
|
||||
<li><a href="#two-step-trick-2" id="toc-two-step-trick-2">Two Step
|
||||
Trick</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
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@ -1605,6 +1607,7 @@ class="sourceCode python"><code class="sourceCode python"><span id="cb6-1"><a hr
|
||||
<div class="sourceCode" id="cb7"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<p></ij></ij></p>
|
||||
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-devroyeRandomSampling1986" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
|
||||
@ -279,6 +279,7 @@ id="toc-acknowledgments">Acknowledgments</a></li>
|
||||
<li><a href="#uncorrelated-disorder-model"
|
||||
id="toc-uncorrelated-disorder-model"><span id="app:disorder_model"
|
||||
label="app:disorder_model"></span> UNCORRELATED DISORDER MODEL</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
{% endcapture %}
|
||||
|
||||
@ -299,6 +300,7 @@ id="toc-acknowledgments">Acknowledgments</a></li>
|
||||
<li><a href="#uncorrelated-disorder-model"
|
||||
id="toc-uncorrelated-disorder-model"><span id="app:disorder_model"
|
||||
label="app:disorder_model"></span> UNCORRELATED DISORDER MODEL</a></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
||||
@ -696,6 +698,7 @@ H_{\mathrm{DM}} = & \;U \sum_{i} (-1)^i \; d_i \;(c^\dag_{i}c_{i} -
|
||||
\nonumber\end{aligned}\]</span></p>
|
||||
<div class="sourceCode" id="cb1"><pre
|
||||
class="sourceCode python"><code class="sourceCode python"></code></pre></div>
|
||||
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-binderFiniteSizeScaling1981" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
|
||||
@ -251,6 +251,7 @@ modes</a></li>
|
||||
<li><a href="#anyonic-statistics" id="toc-anyonic-statistics">Anyonic
|
||||
Statistics</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
{% endcapture %}
|
||||
|
||||
@ -305,6 +306,7 @@ modes</a></li>
|
||||
<li><a href="#anyonic-statistics" id="toc-anyonic-statistics">Anyonic
|
||||
Statistics</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
||||
@ -598,7 +600,7 @@ topologically protected qubits since the four sectors can only be mixed
|
||||
by a highly non-local perturbations <span class="citation"
|
||||
data-cites="kitaevFaulttolerantQuantumComputation2003"> [<a
|
||||
href="#ref-kitaevFaulttolerantQuantumComputation2003"
|
||||
role="doc-biblioref">1</a>]</span>.</p>
|
||||
role="doc-biblioref"><strong>kitaevFaulttolerantQuantumComputation2003?</strong></a>]</span>.</p>
|
||||
<p>Takeaway: The Extended Hilbert Space decomposes into a direct product
|
||||
of Flux Sectors, four Topological Sectors and a set of gauge
|
||||
symmetries.</p>
|
||||
@ -735,7 +737,7 @@ reduces to a determinant of the Q matrix and the fermion parity,
|
||||
see <span class="citation"
|
||||
data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a
|
||||
href="#ref-pedrocchiPhysicalSolutionsKitaev2011"
|
||||
role="doc-biblioref">2</a>]</span>. The only difference from the
|
||||
role="doc-biblioref">1</a>]</span>. The only difference from the
|
||||
honeycomb case is that we cannot explicitly compute the factors <span
|
||||
class="math inline">\(p_x,p_y,p_z = \pm\;1\)</span> that arise from
|
||||
reordering the b operators such that pairs of vertices linked by the
|
||||
@ -761,7 +763,7 @@ determined by fermionic occupation numbers <span
|
||||
class="math inline">\(n_i\)</span>. As discussed in <span
|
||||
class="citation" data-cites="pedrocchiPhysicalSolutionsKitaev2011"> [<a
|
||||
href="#ref-pedrocchiPhysicalSolutionsKitaev2011"
|
||||
role="doc-biblioref">2</a>]</span>, <span
|
||||
role="doc-biblioref">1</a>]</span>, <span
|
||||
class="math inline">\(\hat{\pi}\)</span> is gauge invariant in the sense
|
||||
that <span class="math inline">\([\hat{\pi}, D_i] = 0\)</span>.</p>
|
||||
<p>This implies that <span class="math inline">\(det(Q^u) \prod -i
|
||||
@ -770,7 +772,7 @@ invariant models this quantity which can be related to the parity of the
|
||||
number of vortex pairs in the system <span class="citation"
|
||||
data-cites="yaoAlgebraicSpinLiquid2009"> [<a
|
||||
href="#ref-yaoAlgebraicSpinLiquid2009"
|
||||
role="doc-biblioref">3</a>]</span>.</p>
|
||||
role="doc-biblioref">2</a>]</span>.</p>
|
||||
<p>All these factors take values <span class="math inline">\(\pm
|
||||
1\)</span> so <span class="math inline">\(\mathcal{P}_0\)</span> is 0 or
|
||||
1 for a particular state. Since <span
|
||||
@ -802,9 +804,9 @@ diameters of the torus and, then, annihilating them again.</figcaption>
|
||||
<p>More general arguments <span class="citation"
|
||||
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a
|
||||
href="#ref-chungExplicitMonodromyMoore2007"
|
||||
role="doc-biblioref">4</a>,<a
|
||||
role="doc-biblioref">3</a>,<a
|
||||
href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007"
|
||||
role="doc-biblioref">5</a>]</span> imply that <span
|
||||
role="doc-biblioref">4</a>]</span> imply that <span
|
||||
class="math inline">\(det(Q^u) \prod -i u_{ij}\)</span> has an
|
||||
interesting relationship to the topological fluxes. In the non-Abelian
|
||||
phase, we expect that it will change sign in exactly one of the four
|
||||
@ -894,7 +896,7 @@ definition, the vortex free sector.</p>
|
||||
corresponds to the state where all <span class="math inline">\(u_{jk} =
|
||||
1\)</span>. This implies that the flux free sector is the ground state
|
||||
sector <span class="citation" data-cites="lieb_flux_1994"> [<a
|
||||
href="#ref-lieb_flux_1994" role="doc-biblioref">6</a>]</span>.</p>
|
||||
href="#ref-lieb_flux_1994" role="doc-biblioref">5</a>]</span>.</p>
|
||||
<p>Lieb’s theorem does not generalise easily to the amorphous case.
|
||||
However, we can get some intuition by examining the problem that will
|
||||
lead to a guess for the ground state. We will then provide numerical
|
||||
@ -976,7 +978,7 @@ that form each plaquette and the choice of sign gives a twofold chiral
|
||||
ground state degeneracy.</p>
|
||||
<p>This conjecture is consistent with Lieb’s theorem on regular
|
||||
lattices <span class="citation" data-cites="lieb_flux_1994"> [<a
|
||||
href="#ref-lieb_flux_1994" role="doc-biblioref">6</a>]</span> and is
|
||||
href="#ref-lieb_flux_1994" role="doc-biblioref">5</a>]</span> and is
|
||||
supported by numerical evidence. As noted before, any flux that differs
|
||||
from the ground state is an excitation which we call a vortex.</p>
|
||||
<h3 id="finite-size-effects">Finite size effects</h3>
|
||||
@ -1031,8 +1033,8 @@ of the odd plaquettes does not matter.</p>
|
||||
<p>This happens because we have broken the time reversal symmetry of the
|
||||
original model by adding odd plaquettes <span class="citation"
|
||||
data-cites="Chua2011 yaoExactChiralSpin2007 ChuaPRB2011 Fiete2012 Natori2016 Wu2009 Peri2020 WangHaoranPRB2021"> [<a
|
||||
href="#ref-Chua2011" role="doc-biblioref">7</a>–<a
|
||||
href="#ref-WangHaoranPRB2021" role="doc-biblioref">14</a>]</span>.</p>
|
||||
href="#ref-Chua2011" role="doc-biblioref">6</a>–<a
|
||||
href="#ref-WangHaoranPRB2021" role="doc-biblioref">13</a>]</span>.</p>
|
||||
<p>Similarly to the behaviour of the original Kitaev model in response
|
||||
to a magnetic field, we get two degenerate ground states of different
|
||||
handedness. Practically speaking, one ground state is related to the
|
||||
@ -1040,7 +1042,7 @@ other by inverting the imaginary <span
|
||||
class="math inline">\(\phi\)</span> fluxes <span class="citation"
|
||||
data-cites="yaoExactChiralSpin2007"> [<a
|
||||
href="#ref-yaoExactChiralSpin2007"
|
||||
role="doc-biblioref">8</a>]</span>.</p>
|
||||
role="doc-biblioref">7</a>]</span>.</p>
|
||||
<h2 id="phases-of-the-kitaev-model">Phases of the Kitaev Model</h2>
|
||||
<p>discuss the Abelian A phase / toric code phase / anisotropic
|
||||
phase</p>
|
||||
@ -1167,23 +1169,23 @@ and construct the set <span class="math inline">\((+1, +1), (+1, -1),
|
||||
<figure>
|
||||
<img src="/assets/thesis/amk_chapter/topological_fluxes.png"
|
||||
data-short-caption="Topological Fluxes" style="width:57.0%"
|
||||
alt="Figure 14: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts that both had a jam filling and a hole, this analogy would be a lot easier to make [15]." />
|
||||
alt="Figure 14: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts that both had a jam filling and a hole, this analogy would be a lot easier to make [14]." />
|
||||
<figcaption aria-hidden="true"><span>Figure 14:</span> Wilson loops that
|
||||
wind the major or minor diameters of the torus measure flux winding
|
||||
through the hole of the doughnut/torus or through the filling. If they
|
||||
made doughnuts that both had a jam filling and a hole, this analogy
|
||||
would be a lot easier to make <span class="citation"
|
||||
data-cites="parkerWhyDoesThis"> [<a href="#ref-parkerWhyDoesThis"
|
||||
role="doc-biblioref">15</a>]</span>.</figcaption>
|
||||
role="doc-biblioref">14</a>]</span>.</figcaption>
|
||||
</figure>
|
||||
</div>
|
||||
<p>However, in the non-Abelian phase we have to wrangle with
|
||||
monodromy <span class="citation"
|
||||
data-cites="chungExplicitMonodromyMoore2007 oshikawaTopologicalDegeneracyNonAbelian2007"> [<a
|
||||
href="#ref-chungExplicitMonodromyMoore2007"
|
||||
role="doc-biblioref">4</a>,<a
|
||||
role="doc-biblioref">3</a>,<a
|
||||
href="#ref-oshikawaTopologicalDegeneracyNonAbelian2007"
|
||||
role="doc-biblioref">5</a>]</span>. Monodromy is the behaviour of
|
||||
role="doc-biblioref">4</a>]</span>. Monodromy is the behaviour of
|
||||
objects as they move around a singularity. This manifests here in that
|
||||
the identity of a vortex and cloud of Majoranas can change as we wind
|
||||
them around the torus in such a way that, rather than annihilating to
|
||||
@ -1192,9 +1194,9 @@ ground state. This means that we end up with only three degenerate
|
||||
ground states in the non-Abelian phase <span class="math inline">\((+1,
|
||||
+1), (+1, -1), (-1, +1)\)</span> <span class="citation"
|
||||
data-cites="chungTopologicalQuantumPhase2010 yaoAlgebraicSpinLiquid2009"> [<a
|
||||
href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">3</a>,<a
|
||||
href="#ref-yaoAlgebraicSpinLiquid2009" role="doc-biblioref">2</a>,<a
|
||||
href="#ref-chungTopologicalQuantumPhase2010"
|
||||
role="doc-biblioref">16</a>]</span>. Concretely, this is because the
|
||||
role="doc-biblioref">15</a>]</span>. Concretely, this is because the
|
||||
projector enforces both flux and fermion parity. When we wind a vortex
|
||||
around both non-contractible loops of the torus, it flips the flux
|
||||
parity. Therefore, we have to introduce a fermionic excitation to make
|
||||
@ -1205,24 +1207,17 @@ proposals to use this ground state degeneracy to implement both
|
||||
passively fault tolerant and actively stabilised quantum
|
||||
computations <span class="citation"
|
||||
data-cites="kitaevFaulttolerantQuantumComputation2003 poulinStabilizerFormalismOperator2005 hastingsDynamicallyGeneratedLogical2021"> [<a
|
||||
href="#ref-kitaevFaulttolerantQuantumComputation2003"
|
||||
role="doc-biblioref">1</a>,<a
|
||||
href="#ref-poulinStabilizerFormalismOperator2005"
|
||||
role="doc-biblioref">17</a>,<a
|
||||
role="doc-biblioref">16</a>,<a
|
||||
href="#ref-hastingsDynamicallyGeneratedLogical2021"
|
||||
role="doc-biblioref">18</a>]</span>.</p>
|
||||
role="doc-biblioref">17</a>,<a
|
||||
href="#ref-kitaevFaulttolerantQuantumComputation2003"
|
||||
role="doc-biblioref"><strong>kitaevFaulttolerantQuantumComputation2003?</strong></a>]</span>.</p>
|
||||
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-kitaevFaulttolerantQuantumComputation2003"
|
||||
class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">A.
|
||||
Yu. Kitaev, <em><a
|
||||
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-Tolerant
|
||||
Quantum Computation by Anyons</a></em>, Annals of Physics
|
||||
<strong>303</strong>, 2 (2003).</div>
|
||||
</div>
|
||||
<div id="ref-pedrocchiPhysicalSolutionsKitaev2011" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">F.
|
||||
<div class="csl-left-margin">[1] </div><div class="csl-right-inline">F.
|
||||
L. Pedrocchi, S. Chesi, and D. Loss, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.84.165414">Physical solutions of
|
||||
the Kitaev honeycomb model</a></em>, Phys. Rev. B <strong>84</strong>,
|
||||
@ -1230,7 +1225,7 @@ the Kitaev honeycomb model</a></em>, Phys. Rev. B <strong>84</strong>,
|
||||
</div>
|
||||
<div id="ref-yaoAlgebraicSpinLiquid2009" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">H.
|
||||
<div class="csl-left-margin">[2] </div><div class="csl-right-inline">H.
|
||||
Yao, S.-C. Zhang, and S. A. Kivelson, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevLett.102.217202">Algebraic Spin
|
||||
Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett.
|
||||
@ -1238,7 +1233,7 @@ Liquid in an Exactly Solvable Spin Model</a></em>, Phys. Rev. Lett.
|
||||
</div>
|
||||
<div id="ref-chungExplicitMonodromyMoore2007" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">S.
|
||||
<div class="csl-left-margin">[3] </div><div class="csl-right-inline">S.
|
||||
B. Chung and M. Stone, <em><a
|
||||
href="https://doi.org/10.1088/1751-8113/40/19/001">Explicit Monodromy of
|
||||
Moore–Read Wavefunctions on a Torus</a></em>, J. Phys. A: Math. Theor.
|
||||
@ -1246,20 +1241,20 @@ Moore–Read Wavefunctions on a Torus</a></em>, J. Phys. A: Math. Theor.
|
||||
</div>
|
||||
<div id="ref-oshikawaTopologicalDegeneracyNonAbelian2007"
|
||||
class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">M.
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||||
<div class="csl-left-margin">[4] </div><div class="csl-right-inline">M.
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||||
Oshikawa, Y. B. Kim, K. Shtengel, C. Nayak, and S. Tewari, <em><a
|
||||
href="https://doi.org/10.1016/j.aop.2006.08.001">Topological Degeneracy
|
||||
of Non-Abelian States for Dummies</a></em>, Annals of Physics
|
||||
<strong>322</strong>, 1477 (2007).</div>
|
||||
</div>
|
||||
<div id="ref-lieb_flux_1994" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">E.
|
||||
<div class="csl-left-margin">[5] </div><div class="csl-right-inline">E.
|
||||
H. Lieb, <em><a href="https://doi.org/10.1103/PhysRevLett.73.2158">Flux
|
||||
Phase of the Half-Filled Band</a></em>, Physical Review Letters
|
||||
<strong>73</strong>, 2158 (1994).</div>
|
||||
</div>
|
||||
<div id="ref-Chua2011" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">V.
|
||||
<div class="csl-left-margin">[6] </div><div class="csl-right-inline">V.
|
||||
Chua, H. Yao, and G. A. Fiete, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.83.180412">Exact Chiral Spin
|
||||
Liquid with Stable Spin Fermi Surface on the Kagome Lattice</a></em>,
|
||||
@ -1267,21 +1262,21 @@ Phys. Rev. B <strong>83</strong>, 180412 (2011).</div>
|
||||
</div>
|
||||
<div id="ref-yaoExactChiralSpin2007" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">H.
|
||||
<div class="csl-left-margin">[7] </div><div class="csl-right-inline">H.
|
||||
Yao and S. A. Kivelson, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevLett.99.247203">An Exact Chiral
|
||||
Spin Liquid with Non-Abelian Anyons</a></em>, Phys. Rev. Lett.
|
||||
<strong>99</strong>, 247203 (2007).</div>
|
||||
</div>
|
||||
<div id="ref-ChuaPRB2011" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">V.
|
||||
<div class="csl-left-margin">[8] </div><div class="csl-right-inline">V.
|
||||
Chua and G. A. Fiete, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.84.195129">Exactly Solvable
|
||||
Topological Chiral Spin Liquid with Random Exchange</a></em>, Phys. Rev.
|
||||
B <strong>84</strong>, 195129 (2011).</div>
|
||||
</div>
|
||||
<div id="ref-Fiete2012" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">G.
|
||||
<div class="csl-left-margin">[9] </div><div class="csl-right-inline">G.
|
||||
A. Fiete, V. Chua, M. Kargarian, R. Lundgren, A. Rüegg, J. Wen, and V.
|
||||
Zyuzin, <em><a
|
||||
href="https://doi.org/10.1016/j.physe.2011.11.011">Topological
|
||||
@ -1289,19 +1284,19 @@ Insulators and Quantum Spin Liquids</a></em>, Physica E: Low-Dimensional
|
||||
Systems and Nanostructures <strong>44</strong>, 845 (2012).</div>
|
||||
</div>
|
||||
<div id="ref-Natori2016" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">W.
|
||||
<div class="csl-left-margin">[10] </div><div class="csl-right-inline">W.
|
||||
M. H. Natori, E. C. Andrade, E. Miranda, and R. G. Pereira, <em><a
|
||||
href="https://link.aps.org/doi/10.1103/PhysRevLett.117.017204">Chiral
|
||||
Spin-Orbital Liquids with Nodal Lines</a></em>, Phys. Rev. Lett.
|
||||
<strong>117</strong>, 017204 (2016).</div>
|
||||
</div>
|
||||
<div id="ref-Wu2009" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">C.
|
||||
<div class="csl-left-margin">[11] </div><div class="csl-right-inline">C.
|
||||
Wu, D. Arovas, and H.-H. Hung, <em>Γ-Matrix Generalization of the Kitaev
|
||||
Model</em>, Physical Review B <strong>79</strong>, 134427 (2009).</div>
|
||||
</div>
|
||||
<div id="ref-Peri2020" class="csl-entry" role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">V.
|
||||
<div class="csl-left-margin">[12] </div><div class="csl-right-inline">V.
|
||||
Peri, S. Ok, S. S. Tsirkin, T. Neupert, G. Baskaran, M. Greiter, R.
|
||||
Moessner, and R. Thomale, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.101.041114">Non-Abelian Chiral
|
||||
@ -1310,7 +1305,7 @@ Spin Liquid on a Simple Non-Archimedean Lattice</a></em>, Phys. Rev. B
|
||||
</div>
|
||||
<div id="ref-WangHaoranPRB2021" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">H.
|
||||
<div class="csl-left-margin">[13] </div><div class="csl-right-inline">H.
|
||||
Wang and A. Principi, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.104.214422">Majorana Edge and
|
||||
Corner States in Square and Kagome Quantum Spin-3/2 Liquids</a></em>,
|
||||
@ -1318,14 +1313,14 @@ Phys. Rev. B <strong>104</strong>, 214422 (2021).</div>
|
||||
</div>
|
||||
<div id="ref-parkerWhyDoesThis" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[15] </div><div
|
||||
<div class="csl-left-margin">[14] </div><div
|
||||
class="csl-right-inline"><em><a
|
||||
href="https://www.youtube.com/watch?v=ymF1bp-qrjU">Why Does This Balloon
|
||||
Have -1 Holes?</a></em> (n.d.).</div>
|
||||
</div>
|
||||
<div id="ref-chungTopologicalQuantumPhase2010" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">S.
|
||||
<div class="csl-left-margin">[15] </div><div class="csl-right-inline">S.
|
||||
B. Chung, H. Yao, T. L. Hughes, and E.-A. Kim, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevB.81.060403">Topological Quantum
|
||||
Phase Transition in an Exactly Solvable Model of a Chiral Spin Liquid at
|
||||
@ -1334,7 +1329,7 @@ Finite Temperature</a></em>, Phys. Rev. B <strong>81</strong>, 060403
|
||||
</div>
|
||||
<div id="ref-poulinStabilizerFormalismOperator2005" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">D.
|
||||
<div class="csl-left-margin">[16] </div><div class="csl-right-inline">D.
|
||||
Poulin, <em><a
|
||||
href="https://doi.org/10.1103/PhysRevLett.95.230504">Stabilizer
|
||||
Formalism for Operator Quantum Error Correction</a></em>, Phys. Rev.
|
||||
@ -1342,7 +1337,7 @@ Lett. <strong>95</strong>, 230504 (2005).</div>
|
||||
</div>
|
||||
<div id="ref-hastingsDynamicallyGeneratedLogical2021" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[18] </div><div class="csl-right-inline">M.
|
||||
<div class="csl-left-margin">[17] </div><div class="csl-right-inline">M.
|
||||
B. Hastings and J. Haah, <em><a
|
||||
href="https://doi.org/10.22331/q-2021-10-19-564">Dynamically Generated
|
||||
Logical Qubits</a></em>, Quantum <strong>5</strong>, 564 (2021).</div>
|
||||
|
||||
@ -237,6 +237,7 @@ back from Bond Sectors to the Physical Subspace</a></li>
|
||||
id="toc-open-boundary-conditions">Open boundary conditions</a></li>
|
||||
</ul></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
{% endcapture %}
|
||||
|
||||
@ -277,6 +278,7 @@ back from Bond Sectors to the Physical Subspace</a></li>
|
||||
id="toc-open-boundary-conditions">Open boundary conditions</a></li>
|
||||
</ul></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
||||
@ -905,6 +907,7 @@ which we set to 1 when calculating the projector.</p>
|
||||
anyway, an arbitrary pairing of the unpaired <span
|
||||
class="math inline">\(b^\alpha\)</span> operators could be performed.
|
||||
</i,j></i,j></p>
|
||||
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-marsalTopologicalWeaireThorpe2020" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
|
||||
@ -293,6 +293,7 @@ flux sectors and bond sectors</a></li>
|
||||
<li><a href="#chern-markers" id="toc-chern-markers">Chern
|
||||
Markers</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
{% endcapture %}
|
||||
|
||||
@ -326,6 +327,7 @@ flux sectors and bond sectors</a></li>
|
||||
<li><a href="#chern-markers" id="toc-chern-markers">Chern
|
||||
Markers</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
||||
@ -757,6 +759,7 @@ system.</p>
|
||||
<p><strong>Expand on definition here</strong></p>
|
||||
<p><strong>Discuss link between Chern number and Anyonic
|
||||
Statistics</strong></p>
|
||||
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-mitchellAmorphousTopologicalInsulators2018"
|
||||
class="csl-entry" role="doc-biblioentry">
|
||||
|
||||
@ -241,6 +241,7 @@ Realisations and Signatures</a></li>
|
||||
<li><a href="#generalisations"
|
||||
id="toc-generalisations">Generalisations</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
{% endcapture %}
|
||||
|
||||
@ -285,6 +286,7 @@ Realisations and Signatures</a></li>
|
||||
<li><a href="#generalisations"
|
||||
id="toc-generalisations">Generalisations</a></li>
|
||||
</ul></li>
|
||||
<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
|
||||
</ul>
|
||||
</nav>
|
||||
-->
|
||||
@ -829,6 +831,7 @@ href="#ref-Wu2009" role="doc-biblioref">47</a>]</span></p>
|
||||
quantum many body phases albeit material candidates aplenty. We expect
|
||||
our exact chiral amorphous spin liquid to find many generalisation to
|
||||
realistic amorphous quantum magnets and beyond.</p>
|
||||
<h1 class="unnumbered" id="bibliography">Bibliography</h1>
|
||||
<div id="refs" class="references csl-bib-body" role="doc-bibliography">
|
||||
<div id="ref-kitaevAnyonsExactlySolved2006" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
@ -935,7 +938,7 @@ Conductivity as a Local Chern Marker</a></em>, arXiv Preprint
|
||||
<div id="ref-kitaev_fault-tolerant_2003" class="csl-entry"
|
||||
role="doc-biblioentry">
|
||||
<div class="csl-left-margin">[14] </div><div class="csl-right-inline">A.
|
||||
Y. Kitaev, <em><a
|
||||
Yu. Kitaev, <em><a
|
||||
href="https://doi.org/10.1016/S0003-4916(02)00018-0">Fault-Tolerant
|
||||
Quantum Computation by Anyons</a></em>, Annals of Physics
|
||||
<strong>303</strong>, 2 (2003).</div>
|
||||
|
||||
@ -2,9 +2,8 @@
|
||||
<li>Introduction</li>
|
||||
<ul>
|
||||
<li><a href="./1_Introduction/1_Intro.html#interacting-quantum-many-body-systems">Interacting Quantum Many Body Systems</a></li>
|
||||
<li><a href="./1_Introduction/1_Intro.html#mott-insulators-and-the-hubbard-model">Mott Insulators and The Hubbard Model</a></li>
|
||||
<li><a href="./1_Introduction/1_Intro.html#spin-liquids">Spin Liquids</a></li>
|
||||
<li><a href="./1_Introduction/1_Intro.html#exactly-solvable-models">Exactly Solvable Models</a></li>
|
||||
<li><a href="./1_Introduction/1_Intro.html#mott-insulators">Mott Insulators</a></li>
|
||||
<li><a href="./1_Introduction/1_Intro.html#quantum-spin-liquids">Quantum Spin Liquids</a></li>
|
||||
<li><a href="./1_Introduction/1_Intro.html#outline">Outline</a></li>
|
||||
</ul>
|
||||
<li>Background</li>
|
||||
@ -31,7 +30,6 @@
|
||||
</ul></ul>
|
||||
<li>Chapter 3: The Long Range Falikov-Kimball Model</li>
|
||||
<ul><ul>
|
||||
<li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#contributions">Contributions</a></li>
|
||||
<li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#chapter-summary">Chapter Summary</a></li>
|
||||
</ul>
|
||||
<li><a href="./3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html#the-model">The Model</a></li>
|
||||
|
||||
@ -3,10 +3,10 @@
|
||||
|
||||
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|
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|
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|
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|
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|
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|
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id="tspan543">Quantum</tspan></text>
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id="tspan505">Quantum</tspan></text>
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style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:13.3333px;font-family:'Times New Roman';-inkscape-font-specification:'Times New Roman, Normal';font-variant-ligatures:normal;font-variant-caps:normal;font-variant-numeric:normal;font-variant-east-asian:normal;white-space:pre;shape-inside:url(#rect1816-6);fill:#000000;fill-opacity:1;stroke:none;stroke-width:29.333;stroke-linejoin:round;stroke-opacity:1"
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transform="translate(155.20677,103.19604)"><tspan
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transform="matrix(1.3114339,0,0,1.3114339,281.65359,136.91623)"><tspan
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|
Before Width: | Height: | Size: 11 KiB After Width: | Height: | Size: 12 KiB |
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